The Multi-Agent Path Finding with Task Assignment (MAPF-TA) problem combines task allocation and collision-free path finding for multiple agents within a graph. It can be solved by an extension of the well-known Conflict-Based Search (CBS) algorithm called CBS-TA, which has been demonstrated to be optimal in terms of the sum of costs of all agents. While coordination between agents in MAPF-TA is limited to no-collision constraints, real-world scenarios may require cooperation between agents. For instance, in an exploration mission involving a system of multiple robots, one robot might need to enter a hazardous area only if another agent is able to monitor this area from a support location. Given a hazardous location and its corresponding support location, this coordination requirement can be modeled by a support constraint. In this paper, we propose an extension of the CBS-TA algorithm to handle these support conflicts. In addition, we improve the algorithm’s performance by introducing an alternative cost matrix for task assignment, which takes into account support coordination while maintaining the optimality of the CBS-TA algorithm. We compare the proposed approach to a greedy algorithm in which the task assignment and path finding problems are decoupled, and using two different assignment matrices: the original matrix of CBS-TA and our support-aware matrix. Experiments are carried out using standard MAPF benchmark instances, showing that the proposed cost matrix improves search time and increases the number of instances solved within a given timeout.